Problem: Is ${656823}$ divisible by $9$ ?
Solution: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {656823}= &&{6}\cdot100000+ \\&&{5}\cdot10000+ \\&&{6}\cdot1000+ \\&&{8}\cdot100+ \\&&{2}\cdot10+ \\&&{3}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {656823}= &&{6}(99999+1)+ \\&&{5}(9999+1)+ \\&&{6}(999+1)+ \\&&{8}(99+1)+ \\&&{2}(9+1)+ \\&&{3} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {656823}= &&\gray{6\cdot99999}+ \\&&\gray{5\cdot9999}+ \\&&\gray{6\cdot999}+ \\&&\gray{8\cdot99}+ \\&&\gray{2\cdot9}+ \\&& {6}+{5}+{6}+{8}+{2}+{3} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${656823}$ is divisible by $9$ if ${ 6}+{5}+{6}+{8}+{2}+{3}$ is divisible by $9$ Add the digits of ${656823}$ $ {6}+{5}+{6}+{8}+{2}+{3} = {30} $ If ${30}$ is divisible by $9$ , then ${656823}$ must also be divisible by $9$ ${30}$ is not divisible by $9$, therefore ${656823}$ must not be divisible by $9$.